Uppsala University
Master Thesis
30 credits
Exploring the quark correlator of an
axial-vector with two vector currents
Author:
Di An
Supervisor:
Stefan Leupold
Subject Reader:
Andrzej Kupsc
January 6, 2021
“The whole point of science is that most of it is uncertain. That’s why science is exciting—because we don’t know. Science is all about things we don’t understand. The public, of course, imagines science is just a set of facts. But it’s not. Science is a process of exploring, which is always partial. We explore, and we find out things that we understand. We find out things we thought we understood were wrong. That’s how it makes progress.”
Abstract
Important parts of the hadronic loop contributions to the anomalous mag-netic moment of the muon are related to the quark correlator between an axial-vector current and two axial-vector currents (AVV). The AVV correlator has been studied using different methods for different energy regimes in the past. How-ever, to understand the AVV correlator at intermediate energies is very difficult and limited by the fact that one does not have a reliable theory at the interme-diate energies due to the asymptotic freedom. Therefore the estimation of the AVV correlator at intermediate energies relies on models. Different models can be understood as different ways of interpolating between the very low energies where Chiral Perturbation Theory works well and high energies where pertur-bative QCD works well. In the thesis, we start with a BTT1 construction for the basis of the AVV correlator and show that it can be decomposed into four components with three transversal and one longitudinal form factors. Then we saturate the four form factors by hadronic states using Resonance Chiral Theory (RChT) up to chiral order O(p6). To saturate the AVV correlator we construct the most general Lagrangian including pions (Goldstone bosons), and several vector meson and axial-vector meson multiplets. The model we construct can connect to different phenomenological processes and receives constraints from the latter. We work out the asymptotic behaviors of the transition form factors (TFFs) HiAV(Q2) of an axial-vector meson a decaying into an on-shell vector meson v and an off-shell photon γ∗ using quark counting rules in the Breit frame. The constraints of the a → vγ∗ transitions allow one to reduce the 12 model parameters to 2 free parameters. The decay width of a → γL∗γT fixes
the overall size of the 2 free parameters. Thus, we quantitatively calculate the TFFs HiA(Q21, Q22) of an axial-vector meson decaying into two photons (i.e. a → γ∗γ∗) with 1 free parameter. With the model calculation of HiA(Q21, Q22) at hand, a qualitative discussion about the axial-vector meson pole contribu-tion to the muon magnetic moment is presented. In addicontribu-tion, a dispersive representation is provided for an AV quark correlator sandwiched between a soft one-photon state and the vacuum (i.e. h0|T {Jem, 5µ (x) Jemν (0)}|γsofti). Since
the AV quark correlator is closely related to the AVV correlator, following the dispersive representation, we provide an explanation to the structure of the lon-gitudinal component of the AVV correlator in the chiral limit. The connections to the axial anomaly and to the pion TFF in the chiral limit are clarified. Our explanation is then backed up by our hadronic model calculations.
Popul¨
arvetenskaplig sammanfattning p˚
a svenska
Byggstenarna av v¨arlden best˚ar av partiklar. Partiklar interagerar via fyra funda-mentala krafter: den elektromagnetiska kraften, den svaga kraften, den starka kraften samt gravitationskraften. Standardmodellen ¨ar en kvantf¨altteori som beskriver alla k¨anda Elementarpartiklar och tre av de fyra fundamentala krafterna (gravitation-skraften ¨ar inte inkluderad). Gravitationskraften ¨ar s˚apass svag att man kan bortse dess effekt i de flesta omst¨andigheter, vilket g¨or att standardmodellen fortfarande fungerar v¨aldigt v¨al. Ett s¨att att finna fysik bortom standardmodellen ¨ar att utf¨orligt testa alla aspekter av standardmodellen. Under de senaste ˚aren har en elementarpar-tikel, myonen, dragit uppm¨arksammats av fysiker. Detta beror p˚a att det finns en 3-4 Statistisk signifikans avvikelse mellan standardmodellens ber¨akningar av myonens anomala magnetiska momentet och dess experimentella v¨arde. D¨arav har b˚ade teo-retiker och experimentalister anstr¨angt sig f¨or att unders¨oka myonens anomala mag-netiska moment.
Myonen f˚ar en del av sitt magnetiska moment genom interaktion med en foton via utbyte av subatomiska virtuella partiklar. De virtuella partiklarna kan vara ele-mentarpartiklar, vilket saknar intern struktur, och sammansatta partiklar. De sam-mansatta partiklarna ¨ar hadroner. En hadron ¨ar uppbyggd av tv˚a eller fler kvarkar som h˚alls samman av den starka kraften. Om det finns tv˚a kvarkar i en hadron kallas den f¨or en meson (t.ex. pioner) och om det finns tre kvarkar i en hadron kallas den f¨or en baryon (t.ex. protoner och neutroner). Matematiskt beskriver man infromatio-nen om den interna strukturen av hadroner med hj¨alp av formfaktorer. Det ¨ar dessa formfaktorer som man vill unders¨oka hur de bildar myonens magnetiska moment. Vanligtvis ¨ar formfaktorerna f¨or l˚aga energier sv˚ara att h¨arleda fr˚an kvantkromody-namik. Detta beror p˚a att vid l˚aga och intermedi¨ara energier ¨ar kopplingen mellan kvarkarna s˚apass starka att kvantkromodynamiken ¨ar icke-st¨orbar. Experimentellt ¨
ar datan f¨or formfaktorerna hos n˚agra hadroner (t.ex. axial-vektor) ocks˚a v¨aldigt begr¨ansade. Ett tillv¨agag˚angs¨att f¨or att kringg˚a detta problem ¨ar att f¨orlita sig p˚a modeller f¨or att approximera det magnetiska momentet f¨or n˚agra hadroner vid l˚aga och intermedi¨ara energier.
Vi ˚aterv¨ander till myonens magnetiska moment. Om man vill f¨orb¨attra den teo-retiska noggrannheten av ber¨akningar en aning m˚aste man ¨overv¨aga en ¨ok¨ant sv˚art bidrag: Hadronljus fr˚an ljusspridning (HLbL). HLbL kan t¨ankas som tv˚a fotoners spridningsamplitud d¨ar de tv˚a fotonerna interagerar med varandra genom ett utbyte av virtuella mesoner. Om tv˚a fotoner byter axial-vektormesoner ¨ar hela ber¨akningen av HLbL relaterad till en matematisk storhet kallad axial-vektor-vektor-vektor (AVV) kvarkstr¨omskorrelatorn. AVV-korrelatorn ¨ar ocks˚a parametriserad av formfaktorer vilket ¨ar sv˚ara att direkt ber¨akna. I denna avhandling konstruerar vi en modell f¨or AVV-korrelatorn med hj¨alp av effektivf¨altteori f¨or att approximera formfaktorerna av AVV.korrelatorn vid l˚aga och intermedi¨ara energier. Modellen inkluderar pioner π, vektormesoner v, och axial-vektormesoner a.
bidraget till HLbL. Med v˚ar modell f¨or AVV-korrelatorn kan vi ¨aven unders¨oka tv˚a typer av ¨overg˚angar. I den f¨orsta ¨overg˚angen s¨onderfaller en axial-vektor a till en vektrormeson v och en virtuell foton γ∗. I den andra ¨overg˚angen s¨onderfaller en axial-vektor a till tv˚a virtuella fotoner γ∗. F¨orst˚aelse f¨or de tv˚a ¨overg˚angarna hj¨alper till att avsl¨oja information om de interna konfigurationerna av axial-vektor och vek-tormesonet vid l˚aga energier.
V˚art arbete med AVV-korelatorn kommer inte bara att ge insikter i ber¨akningen av myonens anomala magnetiska moment, utan ¨aven hj¨alpa till med f¨orst˚aelsen av de icke-st¨orbara aspekterna av AVV-korrelatorn.
Acknowledgements
My greatest gratitude goes first to my supervisor, Prof. Stefan Leupold, who expertly guided me through my thesis time. I thank him for his patience with my questions (hopefully they are not too stupid) and for our countless interesting discussions. The most important thing is that with his valuable guidance I can see the beauty of QCD and understand how interesting and exciting the unsolved problems are within the QCD sector of the Standard Model; therefore I’ve decided to become a Ph.D. student under his supervision to contribute my tiny effort to the development of QCD. I also want to thank my colleagues: Nora Salone, Moh Moh Aung, and Dr. Elisabetta Perotti for their assistance. My further thanks are to all the members of the Division of Nuclear Physics for welcoming me into their group and for the funny discussions during the lunches.
I also want to thank my friends Dexin Zhou, Yi Peng, and Huifang You, who colored my life during the hard times. Especially the delicious food you cooked made me forget the darkness of the winter in Sweden and the pandemic of the coronavirus. I must thank my friend William Lindberg for his help with the Swedish summary. Hopefully, I can write a Swedish summary next time on my own when I write my Ph.D. thesis!
Last but not the least, I would like to thank my girlfriend, Shan Yang, for her company and unconditional support. Most importantly, none of this could have happened without my family. They support me to do whatever I want to do in my life.
Contents
1 Introduction 1
1.1 Motivation . . . 1
1.2 Outline of the thesis . . . 6
2 Theory Background 7 2.1 QCD and its symmetries . . . 7
2.1.1 Asymptotic freedom . . . 8
2.1.2 Chiral symmetry . . . 9
2.1.3 Spontaneous symmetry breaking . . . 11
2.2 Introduction to Chiral Perturbation Theory . . . 13
2.2.1 Building blocks and leading-order Lagrangian . . . 13
2.2.2 Corrections beyond leading order . . . 16
2.3 A brief introduction to resonance chiral theory . . . 19
2.3.1 Large Nc limit and its symmetry group . . . 19
2.3.2 The RChT Lagrangian for the saturation of AVV at O(p6) . . 22
3 AVV correlator and related quantities 23 3.1 General decomposition . . . 23
3.2 Transition form factors . . . 26
3.3 Hadronic Light by Light contribution and AVV correlator . . . 29
3.3.1 Operator Product Expansion and the HLbL . . . 29
3.3.2 Axial-vector meson pole contributions to aµ,HLbL . . . 30
3.4 AV correlator with a soft photon . . . 31
4 Methods 35 4.1 Sample calculation . . . 35
4.2 Results for the AVV correlator . . . 40
4.3 Form factors of AVV . . . 41
5 High-energy constraints 43 5.1 a → vγ∗ transition form factors . . . 43
5.2 Decay width of a0 1(1260) → γ ∗γ∗ . . . . 49
5.3 Transition form factors of a → γ∗γ∗ . . . 51
6 Results and discussion 53 6.1 TFFs of a → vγ∗ . . . 55
6.2 TFFs of a0 1(1260), f1(1285) → γ∗γ∗ . . . 58
6.2.1 TFFs of f1(1285) → γ∗γ∗ . . . 61
6.2.2 H4A(Q21, Q22) in two models . . . 63
6.3 Model calculation for the longitudinal component of the AVV correlator 68 6.4 Constraints for the AVV correlator . . . 70
A Appendices 74
A.1 SU(3) algebra . . . 74
A.2 Four components of AVV . . . 75
A.3 Proof for the OPE relation . . . 76
A.4 AV correlator and electromagnetic AVV correlator . . . 79
A.5 Massive spin-1 fields . . . 80
A.6 cuV V 3 mn term as a consistency check . . . 82
A.7 Constraints for parameters . . . 83
A.8 Parameter Count . . . 84
1
Introduction
“Who ordered that?”
Isidor Isaac Rabi
1.1
Motivation
In the 1930s, the physicists thought the universe was merely made of atoms which consist of protons, neutrons, and electrons. Then the muon—a particle that carries the same electric charge and spin as electrons but much heavier, was discovered by Carl D. Anderson and Seth Neddermeyer at Caltech in 1936 [1]. “Who ordered that?”, exclaimed the theorist Isidor I. Rabi when the muon was identified. Ninety years later, physicists are still interested in this particle although its mass and half-life have been measured very accurately. It’s because they believe they can see more through muon’s eye. It is the muon anomalous magnetic moment aµ, that both the
theoretical and experimental physicists spare no effort to figure out.
Every lepton carries a magnetic moment due to its intrinsic spin. The relation between the magnetic moment and spin is simply given by:
~
µ = g q 2m
~
S (1.1)
where ~S := ~σ/2 is the spin operator2 in the spin-12 representation. q and m are the electric charge and mass, respectively. In Fig. 1 we present a graphical representation of the electron magnetic moment and spin. The anti-parallel relation is due to the charge of electron being negative: q = −|e|.
Figure 1: Electron magnetic moment. The blue arrows represent the “rotation” of the electron. According to the right-hand-rule the spin vector points upwards. The electron magnetic moment points downwards due to its negative charge.
The g factor in Eq. (1.1) can be predicted using the Dirac equation [2], and was shown to be precisely 2 but experimental results then showed there is a tiny deviation (anomaly) from the prediction. The anomalous magnetic moment of leptons is defined as al:
al =
gl− 2
2 . (1.2)
Julian Schwinger (Fig. 2), one of the greatest physicists of the twentieth century, showed that the first order quantum electrodynamics (QED) contribution a(QED,1)l to the anomalous magnetic moment is
a(QED,1)l = α 2π,
where α is the fine structure constant. At low energies α is about 1371 .
Figure 2: Julian Schwinger with his “laboratory” in his hands [3].
The Standard Model (SM), a quantum field theory which not only incorporates QED but also weak and strong interactions, has a precise prediction consisting of three parts for aµ:
aSMµ = aQEDµ + aWeakµ + aHadµ (1.3) with aWeakµ and aHadµ being weak and hadronic contributions, respectively. Fig. 3 shows the Feynman diagrams of the respective lowest order contributions. The most trou-blesome contributions are from the hadronic parts due to the non-perturbative nature of low-energy quantum chromodynamics (QCD). This can be seen from the uncer-tainties in the latest estimation [4] for aSM
µ :
aSMµ = 116591823(1)(34)(26) × 10−11 (1.4) where the first uncertainty comes from the electro-weak correction and the last two uncertainties come from the leading and sub-leading order of the hadronic contri-butions. Obviously the hadronic part causes the largest uncertainties of the aSM µ
Figure 3: Respective lowest order contributions from QED, weak and hadronic parts [4].
The up-to-date experimental value of aexp
µ comes from the Brookhaven National
Lab-oratory (BNL) E821 experiment published in 2004:
aexpµ = 116592091(54)(33) × 10−11 (1.5) where the first error is statistical and the second systematic.
One can calculate the difference between aexp
µ and aSMµ : ∆aµ = aexpµ − a SM µ = 268(63)(43) × 10 −11 (1.6) where the first error is from experiment and the second is from theory. One can calculate the discrepancy in terms of standard deviations from the aSM
µ [4]:
Z = a
exp
µ − aSMµ
q
(∆aexpµ )2+ (∆aSMµ )2
(1.7)
with ∆aexpµ and ∆aSMµ defined as the experimental and theoretical uncertainties3 re-spectively. The value of Z measures the discrepancy between aexp
µ and aSMµ in terms of
the standard deviation σ. Plugging Eq. (1.6) in Eq. (1.7), we get a 3.5σ deviation [4], which has a strong hint for the existence of beyond standard model physics. Fig. 4 shows a compilation of the recent theory and experimental results for aµ. One can
see both from Fig. 4 that the uncertainties ∆aexp
µ and ∆aSMµ are commensurate.
The new experiment at Fermilab will improve the experimental accuracy for four-fold [5], which will reduce the uncertainty from 63 × 10−11 to about 16 × 10−11. If the central value of the experimental result is unchanged and without further reduction of theoretical uncertainties, a 5σ deviation can be reached. However, it’s also possible that aexp
µ can move towards aSMµ . Therefore on the theory side, one must reduce the
SM uncertainties as much as possible accordingly.
Figure 4: Compilation of the latest results for aµ [4].
As we have pointed out, on the theoretical side, the precision of prediction is limited by the hadronic contributions. The two dominant hadronic contributions are Hadronic Vacuum Polarization (HVP)4 and Hadronic Light by Light scattering (HLbL) (Fig. 5). Numerically, the largest hadronic contribution to aµis HVP, which is of order5O(α2).
Figure 5: Hadronic Light by Light scattering [4].
The second-largest hadronic contribution is the HLbL at O(α3). As we have
ex-plained, one cannot resort to perturbation theory to calculate the hadronic contribu-tions, so one must use either a data-driven method or lattice QCD [:2020ynm, 6]. Although a lot of important progress in lattice QCD has been made in recent years, the uncertainties of its current predictions for hadronic effect are much larger than that of data-driven methods [:2020ynm]. The HVP contribution contains a two-point function of two electromagnetic currents. Luckily, using dispersion theory the
4See the fourth diagram in Fig. 3.
5The HVP contribution starts at order O(α2). The higher order HVP contributions are of the
two-point function can be reconstructed from the cross-section of e+e− → hadrons.
Therefore, the uncertainty of HVP mainly comes from the experimental measurement of the hadronic cross-section. In the future, with a wealth of better accurately mea-sured data the uncertainty of HVP can be reduced [:2020ynm]. Compared to HVP, the HLbL contribution is much smaller but it is believed that the HLbL contribution will soon dominate the theoretical uncertainty. Therefore the g − 2 community has put a huge effort into the study of these hadronic effects. In the past 20 years, the estimates of the HLbL contribution depended on models6[6] due to the breakdown of
perturbation theory at low energies and the lack of direct experimental information. However, one can split HLbL into single-particle pole and many-particle exchange terms. The lower the total mass of the intermediate states, the larger the contri-butions to aµ is, so the largest contribution to HLbL comes from the pion pole [6].
Recently, a data-driven estimation of the pion pole contribution has been published, which is the first time that a model-independent analysis with fully controlled uncer-tainty estimates has ever been made [9].
This work addresses the axial-vector—vector—vector (AVV) quark correlator. The AVV quark corretalor is a time-ordered product of one axial-vector current and two vector currents. The AVV quark correlator will be rigorously defined in Chapter 3. There are two ways how HLbL relates to the AVV correlator. Firstly, If one of the momenta of the photon legs is significantly smaller than the others, the HLbL tensor can be reduced to the AVV correlator using the operator product expansion (OPE) [7]. Secondly, the AVV correlator can be related to the axial-vector meson single-particle pole contribution to HLbL7.
Another reason why we are interested in the AVV correlator is from the theoreti-cal point of view. We know the longitudinal component of the AVV correlator is completely fixed by the chiral anomaly [10], but it is not clear how the longitudinal component emerges. In the thesis we have shown that we can understand the longi-tudinal component from a dispersive point of view and our explanation is backed up by our model calculations.
Although the best ways to determine the hadronic SM contributions to g − 2 is us-ing lattice QCD or data-driven methods based on dispersion theory, for the study of the AVV correlator, both of the two methods fail to apply at the moment. Hence, one needs to resort to phenomenological models. However, by using phenomenologi-cal models, one usually introduces many parameters to paramatrize the Lagrangian. Therefore one needs to put constraints on the parameters as much as possible. The more fundamental constraints one can find for such models, the better those predic-tions get in the sense that they are more realistic. Such constraints come from the regions where we understand QCD best. This is at high energies for quantities where perturbative QCD and/or the OPE can be used and at low energies where Chiral Perturbation Theory can be used. The task of the thesis is to analyze the AVV
cor-6Some models are discussed in [7] [8].
7Schematically, the axial-vector meson contribution is related to h0|Jµ
emJemν |ai whose form factors
relator and related quantities in the same spirit as one analyzes the VVVV correlator of HLbL.
1.2
Outline of the thesis
In chapter 2 we introduce the theoretical concepts and tools relevant to our work. The important concepts include asymptotic freedom, symmetries and approximate symmetries of QCD, spontaneous symmetry breaking. The main model we use in the thesis is the Resonance Chiral Theory (RChT) to be discussed in section 2.3. In subsection 2.3.2 we construct the relevant Lagrangian terms that we use to saturate the form factors of the AVV correlator.
In chapter 3 we define the flavor AVV correlator and construct a basis for it using the BTT construction. Based on the basis which consists of four Lorentz structures, the AVV correlator can be decomposed into four form factors Hi. We then establish
the relations between the form factors Hi and the following two kinds of transition
form factors in section 3.2:
1) The transition form factors HAV
i (Q21, Q22) of an axial-vector meson a decaying to
v and γ∗, where v refers to an on-shell vector-meson and γ∗ refers to an off-shell photon.
2) The transition form factors HA
i (Q21, Q22) of an axial-vector meson a decaying to two
off-shell photons γ∗γ∗.
In section 3.3, we also show how one can connect the HLbL to the flavor AVV cor-relator using an OPE relation. More detailed information about the OPE relation is shown in appendix A.3 for reference.
In section 3.4 we discuss the longitudinal component of AVV and its relation to the chiral anomaly from a dispersive point of view.
In chapter 4 we first give a sample calculation of the AVV correlator using our hadronic model and then the full results of the form factors of the AVV correlator are pre-sented.
In chapter 5 we discuss the details of how to put high-energy constraints on the model parameters.
2
Theory Background
“A theory with mathematical beauty is more likely to be correct than an ugly one that fits some experimental data.”
Paul A. M. Dirac
2.1
QCD and its symmetries
Quantum chromodynamics is a quantum field theory which describes the strong in-teraction between quarks and gluons. It has an SU (3)c symmetry where c stands for
the color quantum number. The Lagrangian of the quantum field theory is given as the following [2, 11]: LQCD = X f =u,d,s,c,b,t ¯ ψf(i /D − mf)ψf − 1 2Tr(GµνG µν) (2.1)
where for each quark flavor f , ψf is a triplet consisting of three color states, red (r),
green (g) and blue (b),
ψf = ψf,r ψf,g ψf,b (2.2)
and we omit the spin indices of quarks. The gauge covariant derivative is defined as Dµ := ∂µ− ig 8 X a=1 λa 2 Aµ,a =: ∂µ− igAµ, (2.3)
where the eight λa
2 matrices are the generators of the SU (3) group, with λa defined
in appendix A.1. The gauge covariant derivative contains eight gluon fields Aµ,a
and the QCD coupling constant g. On the second line we introduce a 3 × 3 matrix Aµ :=P8
a=1 λa
2 Aµ,aso that the covariant derivative can be written in a more compact
form.
Like for the QED case in [12], the field-strength color matrix Gµν can be defined as
[11]
Gµν :=
i
g[Dµ, Dν] . (2.4)
Plugging Eq. (2.3) in Eq. (2.4), it’s straightforward to show that Gµν = Gaµνλ2a with
the gluon field-strength tensor Ga
µν defined as [13]
Gaµν := ∂µAaν − ∂νAaµ− gfabcAbµA c
which leads to gluon self-interactions. Note that the coefficient 1
2 in front of Tr(GµνG µν)
in Eq. (2.1) is to make sure that the kinetic terms of the gluon fields are properly nor-malized.
The form of the Lagrangian in Eq. (2.1) remains unchanged under SU (3)clocal
trans-formations. The quark fields ψf transform in the fundamental representation of
SU (3)c: ψf(x) → U (x)ψf(x) , ¯ ψf(x) → ¯ψf(x)U†(x) (2.6) with U (x) ∈ SU (3)c.
As ¯ψf(i /D − mf)ψf must be invariant under SU (3)c gauge transformation, it means
that Dµψf must transform the same as the quark fields
Dµψf(x) → U (x)Dµψf(x) . (2.7)
From the above relation, one can further derive the transformation property of the gluon fields: Aµ(x) → U (x)(Aµ(x) + i g∂ µ x)U † (x) . (2.8)
We next show that Tr(GµνGµν) is also invariant under SU (3)ctransformation. To see
this we need to first understand how the gauge covariant derivative Dµ transforms.
Since the covariant derivative Dµ contains a gluon field-strength matrix Aµ, plugging
Eq. (2.8) in Eq. (2.3) one can immediate derive
Dµ→ U DµU†. (2.9)
The above equation means
Gµν = i g[Dµ, Dν] SU (3)c −−−−→i gU (x)[Dµ, Dν]U † (x) =U (x)GµνU†(x). (2.10)
So Tr(GµνGµν) is invariant under SU (3)c. In summary, the QCD Lagrangian is
constructed by requiring SU (3)c local gauge symmetry. In the section 2.1.2, we will
discuss some other symmetries of the QCD Lagrangian and their consequences.
2.1.1 Asymptotic freedom
The coupling “constant” g defined in Eq. (2.3) is not a constant. The variation of g is encoded in the renormalization group equation [2, 12]:
dg(µ)
For QCD, we have an SU (3)c gauge group and at high energies 6 active flavors of
quarks. The 1-loop approximation of the beta function β(g) is given as [2] β(g)1-loop = g3 16π2(− 11 3 Nc+ 2 3Nf) = g 3 16π2(− 11 3 × 3 + 2 3 × 6). (2.12)
It is obvious that the above beta function is negative so g will decrease as the energy scale µ increases. We can measure g(µ0) or equivalently αs(µ0) = g(µ0)2/4π at some
physical scale µ0, for instance, αs(µ0 = MZ0) = 0.12 [4]. Once one has fixed g(µ) at a
given scale, the renormalization group equation will predict the value of g(µ) at any other scale. Plugging β1-loop in the renormalization group equation gives the following
solution for αs(µ) [13, 2]: αs(µ) = αs(µ0) 1 + 1 4π( 11 3 Nc− 2 3Nf)log(µ 2/µ2 0) . (2.13)
As µ approaches infinity, αs(µ) → 0, which is called asymptotic freedom. The
asymp-totic freedom is a special feature of QCD. As µ decreases, αs(µ) increases so there is
an energy boundary, ΛQCD, below which αs(µ) is large so that perturbation theory
fails. The energy boundary ΛQCD is typically called the dynamical scale of QCD and
its range is about 0.3-1 GeV [4], meaning that for energies below ΛQCD the theory is
non-perturbative.
In the next subsection, we will introduce more symmetries as they play an important role to help us understand the non-perturbative nature of QCD.
2.1.2 Chiral symmetry
Apart from the SU (3)c gauge symmetry, the QCD Lagrangian of Eq. (2.1) also
pos-sesses some approximate symmetries. The chiral symmetry plays a fundamental role in this thesis and it’s also the basis of Chiral Perturbation Theory (ChPT) and Res-onance Chiral Theory (RChT).
If one determines the masses of quarks [4], one will find that the masses of up, down and strange quarks8 are much lighter than the dynamical scale Λ
QCD:
mu ≈ 0.0022 GeV, md≈ 0.0047 GeV, ms ≈ 0.093 GeV ,
while the masses of the other three quark flavors9,
mc≈ 1.27 GeV, mb ≈ 4.18 GeV, mt ≈ 173 GeV
8Single quarks have never been observed in nature due to color confinement. Therefore the masses
of quarks are not observable and dependent on the renormalization scheme and the energy scale µ. The masses of u, d, s are estimated in the ¯MS scheme at a scale µ ≈ 2 GeV.
9m¯
Q(µ) is determined at the scale µ = ¯mQ for Q = c, b. The usual explanation for mass
measurement of the top quarks is that the top quarks are so heavy that they decay before they combine to form hadrons so in this sense they are “almost” free particles, which allows one to directly determine its mass. The top mass given here is from the direct measurements [4].
are heavier than ΛQCD. This motivates us to consider a limit where the up, down,
and strange quarks are massless, which is called chiral limit and the corresponding symmetry of the QCD Lagrangian in this limit is called chiral symmetry [14]. We note that chiral symmetry is not an exact symmetry in the real world, but an approximate symmetry and we will see that it will be explicitly broken by quark masses and the chiral anomaly. In the chiral limit, we set the masses of light quarks to zero, so Eq. (2.1) becomes: L0,QCD= X f =c,b,t ¯ ψf(i /D − mf)ψf − 1 4G a µνG µν a + ¯qi /Dq (2.14)
where we define q as a collection of the three lightest quarks in a flavor triplet:
q = u d s . (2.15)
In the chiral limit in which the masses of light quarks vanish, the classical version of Eq. (2.14) exhibits a global SU (3)L× SU (3)R × U (1)V × U (1)A symmetry. To
be concrete about what the transformation means, we first define the projection operators PR:= 1 2(1 + γ5) , PL:= 1 2(1 − γ5) . (2.16)
It is obvious that PR + PL = 1. Therefore we can write any fermion state ψ =
PRψ + PLψ =: ψR+ ψL.
The transformation SU (3)L× SU (3)R is defined as
qR → URqR, qR := PRq ,
qL → ULqL, qL := PLq ,
(2.17)
with UL and UR ∈ SU (3). In the chiral limit mu,d,s→ 0, one can show [11] that the
above transformation leads to the following conserved currents: jR,aµ (x) = ¯qR(x)γµ λa 2 qR(x) , jL,aµ (x) = ¯qL(x)γµ λa 2 qL(x) . (2.18)
One should note that the labels λa are flavor indices, not color indices, because UR
and UL are in the SU (3) flavor space.
Then we can define the vector current jV,aµ (x) and axial-vector current jA,aµ (x) as the combinations of the left- and right-handed currents:
jV,aµ (x) = jR,aµ (x) + jL,aµ (x) = ¯q(x)γµλa 2 q(x) , jA,aµ (x) = jR,aµ (x) − jL,aµ (x) = ¯q(x)γµγ5
λa
2 q(x) .
One can show that the transformation Eq. (2.17) is equivalent to the following [11]: q → ei(θαλα/2+βαλαγ5/2)q, (2.20)
with θαand βαbeing the parameters controlling SU (3)V and SU (3)Atransformations,
respectively, meaning that
SU (3)L× SU (3)R = SU (3)V × SU (3)A.
The U (1)V transformation10 is given as q → eiθq and the U (1)A transformation is
given as q → eiθγ5q. The corresponding Noether current for the U (1)
V symmetry
is
Jµ= ¯qγµq (2.21)
and for the U (1)A symmetry, the current is
J5µ= ¯qγµγ5q . (2.22)
Note that in QCD U (1)V is not an approximate symmetry but an exact symmetry
even if one restores all the quark masses. However, U (1)Ais broken in the chiral limit
due to a non-trivial Jacobian in the path integral, which is called chiral anomaly [12].
When restoring the quark mass term, SU (3)V × U (1)V is still a symmetry of QCD
if all three quarks have the same mass, but the SU (3)A symmetry is broken. The
charges which emerge from the currents Eq. (2.19) are defined as:
QVa := Z d3r jV,a0 , a = 1, 2, · · · , 8 QAa := Z d3r jA,a0 , a = 1, 2, · · · , 8. (2.23)
The above charges are conserved only in the chiral limit.
2.1.3 Spontaneous symmetry breaking
In the chiral limit, the vacuum has been proved to be invariant under SU (3)V ×
U (1)V [15], so is the Hamiltonian. Although the Lagrangian possesses an SU (3)A
symmetry, the ground state does not, which is known as spontaneous symmetry breaking. A non-vanishing chiral condensate h¯qqi implies spontaneous breaking of chiral symmetry
h¯qqi 6= 0 ⇒ QAa|0i 6= 0 . (2.24) To prove the above relation, one needs to introduce the operator
Pb = ¯qiγ5
λb
2q(x), b = 1, 2, · · · , 8 . (2.25)
One can show that
[QAa, Pb] = −
i
2δa,bq(x)q(x) .¯ (2.26)
If we sandwich the above equation with the QCD vacuum:
h0|QA
aPb− PbQAa|0i = −
i
2δa,bh¯q(x)q(x)i . (2.27) Hence, QA
a |0i 6= 0 if h¯q(x)q(x)i 6= 0. However, QAa |0i 6= 0 does not necessarily mean
h¯q(x)q(x)i 6= 0 because the operator QAaPb−PbQAa may vanish. The quark condensate
is the sum of up-, down-, and strange-quark condensates:
h¯qqi = h¯uui + h ¯ddi + h¯ssi . (2.28) Obviously, the above equation is invariant under Lorentz and U (3)V transformation.
The latter symmetry implies: 1
3h¯qqi = h¯uui = h ¯ddi = h¯ssi . (2.29) It can be confirmed from lattice QCD, for instance in [16] [17], that the quark con-densates h¯uui ≈ h ¯ddi ≈ h¯ssi are about −(250 MeV)3 in the chiral limit. Therefore,
the non-zero quark condensate implies that the SU (3)A symmetry is spontaneously
broken in the chiral limit11. Many other pieces of evidence can also support this. If the chiral symmetry is not broken, then one expects to see parity doublets, in other words, particles with opposite parity having similar masses. The parity doublet does not exist in nature. For example, pions are the lightest pseudoscalars with masses of about 140 MeV but the lightest scalar meson is f0(500) whose mass is about 500
MeV [4]. The fact that pions are so much lighter than the other hadrons can actu-ally be explained by the spontaneous symmetry breaking of chiral symmetry. One consequence of the spontaneous symmetry breaking is the appearance of massless Goldstone bosons [12]. The number of the Goldstone bosons will be the number of the generators of the broken group. In the case of SU (3)A, this number is 32− 1 = 8.
As we know that chiral symmetry is only an approximate symmetry in nature, the Goldstone bosons will be massive instead.
Again, according to the Goldstone theorem, the quantum numbers of the Goldstone bosons are the same as QA
a |0i. As QAa |0i does not carry any Lorentz indices, the
Goldstone bosons must be scalar or pseudoscalar particles. To see the Goldstone bosons are odd under parity transformation [19, 14]:
P QAaP−1P |0i = −QAa |0i , a = 1, . . . , 8 . (2.30) Hence, the Goldstone bosons are pseudoscalar mesons. They are identified with the 8 lightest pseudoscalars π0, π±, K±, ¯K0, K0 and η.
11From Chiral Perturbation Theory which we will introduce later, one can also relate the quark
condensates to the non-zero pion mass via the GMOR relation [18] worked out by Gell-Mann, Oakes, and Renner in 1969.
2.2
Introduction to Chiral Perturbation Theory
2.2.1 Building blocks and leading-order LagrangianDue to the asymptotic freedom, the coupling constant αs of QCD becomes large at
low energies (E ΛQCD) where the non-perturbative effects take over. We want
to establish a theory at low energies where perturbative QCD fails. There are so far two rigorous approaches to tackle low energy QCD: Chiral Perturbation Theory and lattice QCD. In this section, we introduce the basics of Chiral Perturbation Theory.
Chiral Perturbation Theory is an effective field theory for the eight Goldstone bosons in the energy regime where no other degrees of freedom can be excited. There are two reasons why one can construct such an effective field theory. The first reason is that the Goldstone bosons are very light thanks to spontaneous symmetry breaking, so m2π
m2 hadrons
is12 very small. Thus there exists a mass gap separating the pions from
the rest of the hadronic particles. The second reason is that we can find the right symmetry requirement for the effective field theory Lagrangian containing the eight Goldstone bosons as the only degrees of freedom. Therefore one only needs to find the most general Lagrangian satisfying the symmetry requirement. In this section, we are going to explain how one can realize the symmetry. The logic of the introduction in this section is taken from the book [14].
We want the constructed effective Lagrangian to be invariant under G = SU (3)L×
SU (3)R×U (1)V. Because QAa transforms as an octet under the subgroup H = SU (3)V
the eight pseudoscalar degrees of freedom should also transform as an octet. The ground state of the effective theory should also be invariant under SU (3)V × U (1)V
motivated by spontaneous symmetry breaking. One can regard the Goldstone boson fields as an element of the quotient G/H [14], the set of all the left cosets. We can parametrize the Goldstone boson fields belonging to the coset space G/H as the following [14]: u(φ) = exp(√i 2 Φ F0 ) (2.31) where Φ := √1 2 P8
i=1λiφi with φi being the eight Goldstone bosons. The constant F0
is called pion decay constant in the chiral limit. To be explicit:
Φ = √1 2λiφi = √1 2 φ3+√13φ8 φ1− iφ2 φ4− iφ5 φ1+ iφ2 −φ3+√13φ8 φ6− iφ7 φ4+ iφ5 φ6+ iφ7 −2√3φ8 . (2.32)
One can define φ3(φ8) to be the physical π0(η) fields because they have the respective
isospin and strangeness quantum numbers. Similarly, one can relate the Cartesian
12m
fields φiand the physical fields by the isospin and strangeness quantum numbers: φ1− iφ2 = √ 2π+, φ1+ iφ2 = √ 2π−, φ4− iφ5 = √ 2K+, φ4+ iφ5 = √ 2K−, (2.33) φ6− iφ7 = √ 2K0, φ6+ iφ7 = √ 2 ¯K0. We can plug the physical fields back into Eq. (2.32) to get:
Φ = √1 2 π0+ √1 3η √ 2π+ √2K+ √ 2π− −π0+ √1 3η √ 2K0 √ 2K− √2 ¯K0 √−2 3η . (2.34)
To discuss the transformation properties of u(φ) under the group G, we need to introduce some mathematical concepts. The action of the group G = SU (3)L ×
SU (3)R on u(φ) is given as [20]:
u(φ)−−−−−−−−−−→ gSU (3)L× SU (3)R Ru(φ)h(g, φ)−1 = h(g, φ)u(φ)g †
L (2.35)
where gR, gL13 belong to SU (3)R and SU (3)L, respectively. h(g, φ) ∈ H is a
com-pensator field, and g is an element of the group G, so h(g, φ) depends on the group element g and φ.14 From Eq. (2.35), it’s obvious that U := u(φ)2 should transform as [14]
U −−−−−−−−−−→ USU (3)L× SU (3)R 0 = gRU g †
L. (2.36)
Moreover, to construct the chiral Lagrangian, it’s very useful for us to consider tensors X (to be specified in Eq. (2.38)) which transform as:
X −−−−−−−−−−→ h(g, φ)Xh(g, φ)SU (3)L× SU (3)R †. (2.37) The following tensors transform according to Eq. (2.37):
uµ = u†µ := i[u †(∂ µ− irµ)u − u(∂µ− ilµ)u†] , χ± := u†χu ± uχ†u , f±µν := uFLµνu †± u† FRµνu , hµν := ∇µuν+ ∇νuµ (2.38)
with χ = 2B0(s+ip), where s and p stand for scalar and pseudoscalar external sources
and B0 is a low-energy constant, B0 ≈ 1 GeV. FRµν and F µν
L are defined as:
FRµν := ∂µrν − ∂νrµ− i[r µ, rν] , FLµν := ∂µlν− ∂νlµ− i[l µ, lν] . (2.39) 13g
R and gL are spacetime-dependent, namely: gR(x) and gL(x) 14The explicit form of h(g, φ) is not important.
The covariant derivative ∇µ acting on a tensor X is defined by
∇µX := ∂µX + [Γµ, X] , (2.40)
where the Γµ is the chiral connection:
Γµ:=
1 2{u
†
(∂µ− irµ)u + u(∂µ− ilµ)u†} . (2.41)
We can define a chiral gauge covariant derivative Dµ by its action on any object A
transforming as A → gRAg † L:
DµA := ∂µA − irµA + iAlµ. (2.42)
It’s useful to introduce a vector source vµ and an axial-vector source aµ. They are related to the right- and left-handed sources by:
vµ = 1 2(r µ + lµ) , aµ = 1 2(r µ− lµ) . (2.43)
With all our building blocks in Eq. (2.38) we can build infinitely many terms compat-ible with the underlying symmetries. A natural question is: what is the hierarchy of the infinitely many terms in the Lagrangian? In Chiral Perturbation Theory we have nothing like a coupling constant α in QED or αs in perturbative QCD that allows us
to do a Taylor expansion. That means we need another power counting scheme to organize the Lagrangian. We notice that the derivatives in the Lagrangian translates to momenta:
Dµ∼ O(p) (2.44)
where p denotes a typical energy or momentum which is on the order of the masses of Goldstone bosons in the low energy region. Thus, we aim at a Taylor expansion in powers of p, which should work as long as one stays at low enough energies. Since the definition of the covariant derivative contains left- and right-handed vector fields, lµ and rµ, it should be consistent to require lµ and rµ ∼ O(p). Therefore the field
strengths have the order:
FR/Lµν ∼ O(p2) . (2.45)
We also assume:
U ∼ O(p0) , χ± ∼ O(p2) .
(2.46)
After we figured out the orders of the building blocks, we can write the most general Lagrangian in the leading order compatible with chiral symmetry and parity15:
L(2)eff = 1 4F
2
0huµuµ+ χ+i , (2.47)
15QCD conserves the parity, so the parity of the ChPT Lagrangian must be conserved. By
in terms of U := u(φ)2:
L(2)eff = 1 4F
2
0hDµU (DµU )†+ χU†+ U χ†i (2.48)
where the superscript L(2n)eff in the Lagrangian indicates its chiral order. Due to Lorentz invariance, only even orders are possible.
2.2.2 Corrections beyond leading order
It is possible to work out the most general Lagrangian at O(p4). Such SU (3) L ×
SU (3)Rinvariant Lagrangian was first constructed by Gasser and Leutwyler [21]:
L(4)eff = L1hDµU (DµU )†i2+ L2hDµU (DνU )†ihDµU (DνU )†i
+L3hDµU (DµU )†DνU (DνU )†i + L4hDµU (DµU )†ihχU†+ U χ†i
+L5hDµU (DµU )†(χU†+ U χ†)i + L6hχU†+ U χ†i2
+L7hχU†− U χ†i2+ L8hU χ†U χ†+ χU†χU†i
−iL9hFµνRD µ U (DνU )†+ FµνL(DµU )†(DνU )i + L10hU FµνLU † FRµνi , (2.49)
where L1-L10 are low-energy constants (LECs).
We can always go to higher-order terms to improve the precision of calculation by accounting for loops and including more derivatives in the Lagrangian. The price we have to pay by going to higher-order terms is introducing more low-energy constants that need to be determined.
There are processes that cannot be described by the L(2)eff and L(4)eff terms. For instance, π0 → γγ is not covered in L(2)
eff and L (4)
eff alone, because the decay π0 → γγ must involve
an epsilon tensor in the effective Lagrangian ∼ π0FµνFρσµνρσ, which does not exist
in L(2)eff and L(4)eff. Moreover, K+K−→ π+π−π0 cannot be described by L(2)
eff and L (4) eff.
More generally, interactions involving an odd number of Goldstone bosons are not allowed by L(2)eff and L(4)eff .
To see this, let’s first have a look at the symmetries of Eq. (2.48) and Eq. (2.49). They have three basic discrete symmetries:
1. Charge conjugation: U (t, ~x) → UT(t, ~x)
2. “Space-time” parity P0: U (t, ~x) → U (t, −~x)
3. U (t, ~x) → U†(t, ~x) .
In the charge conjugation symmetry we replace every Goldstone boson to its anti-particle: Φ = √1 2 π0+√1 3η √ 2π+ √2K+ √ 2π− −π0+√1 3η √ 2K0 √ 2K− √2 ¯K0 −2√ 3η → 1 √ 2 π0+ √1 3η √ 2π− √2K− √ 2π+ −π0+ √1 3η √ 2 ¯K0 √ 2K+ √2K0 √−2 3η . (2.50)
It’s easy to see that this amounts to change U to UT. In the “space-time” parity
we simply change the space-time dependence of the fields, which has nothing to do with the true parity of the fields. In the third symmetry, U → U† means φa→ −φa,
which means when we turn off the external fields and Taylor expand U (x), each vertex can only contain an even number of fields, so K+K− → π+π−π0 cannot be
described by L(2)eff and L(4)eff. Going to O(p6) [22] in Chiral Perturbation Theory allows π0 → γγ and K+K− → π+π−π0, but they would be much suppressed. Hence, we
conclude that there is another term that must be missing at O(p4), which is the
famous Wess-Zumino-Witten (WZW) term [23, 24]. The WZW term reproduces the chiral anomaly of QCD at low energies and provides a correct prediction for the decay rate of π0 → γγ.
Witten showed [24] that the action which can reproduce the chiral anomaly at low energies does not exist in Minkowski space, but it exists in the five-dimensional space D where the usual four-dimensional space lives on the boundary ∂D of such a five-dimensional ball. This is a mathematical trick16 to extend the definition of the range of the Goldstone boson fields to a fifth dimension. The WZW action is given as:
SW ZW = k
Z
D
d5y ω, (2.51)
where ω is defined as: ω = − i 240π2 µνρστ Tr(U†∂U ∂yµU †∂U ∂yνU †∂U ∂yρU †∂U ∂yσU †∂U ∂yτ) (2.52)
and the U (y) field is defined on the five-dimensional space D,
U (y) = exp(iαπ(y) F0
) ,
y = (xµ, α), (0 ≤ α ≤ 1)
(2.53)
where y are the space-time coordinates in the five-dimensional space D. Our Minkowski space is defined on the surface of the five-dimensional space for α = 1. It can also be shown that k = Nc= 3 [14].
Now let’s analyze the symmetry of Eq. (2.51). First we note that Eq. (2.51) is invariant under the transformation U → gRU g
†
L17, which is a basic symmetry requirement from
QCD. Next, we note that under the charge conjugation U → UT, ω is still invariant because of the trace. However, the “space-time” parity and U → U† symmetry are broken in WZW. Due to the presence of an extra fifth dimension, the breaking of the two symmetries is not easy to see. We can first expand U in terms of the Goldstone bosons: U = exp( i F0 λaπa) = 1 + i F0 λaπa+ O((λaπa)2) , (2.54)
16Mathematically the trick used by Witten is motivated by the work related to Dirac magnetic
monopoles [24].
17We assume the action of G is global. We will make the WZW action local gauge invariant later
yielding:
U†∂µU ≈
i F0
∂µ(λaπa) . (2.55)
Plugging the above equation back into Eq. (2.51) gives: Z D d5y ω = 2 15π2F5 0 Z D d5y µνρστtr(∂µπ∂νπ∂ρπ∂σπ∂τπ) + O(π6) = 2 15π2F5 0 Z D d5y µνρστ∂µtr(π∂νπ∂ρπ∂σπ∂τπ) + O(π6) . (2.56)
On the second line we’ve used the fact that ∂µ∂ν is symmetric so the anti-symmetric
tensor multiplying ∂µ∂ν is 0. We notice that now the integrand contains a partial
derivative. We can use Stokes’ theorem to rewrite the integral into an integral on the boundary ∂D which by construction, is our space-time [24]:
Z D d5y ω = 2 15π2F5 0 Z D d5y µνρστ∂µtr(π∂νπ∂ρπ∂σπ∂τπ) + O(π6) = 2 15π2F5 0 Z space−time d4x νρστtr(π∂νπ∂ρπ∂σπ∂τπ) + O(π6) . (2.57)
Now the leading term in the above equation does not contain the “space-time” sym-metry P0. To see that we first note that under P0:
π∂νπ∂ρπ∂σπ∂τπ → π∂νπ∂ρπ∂σπ∂τπ , (2.58)
and we can rewrite the Levi-Civita tensor as
νρστ = −νρστ, (2.59)
meaning that the leading term of the WZW action transforms into its negative under P0. Further more, there is an odd number of Goldstone bosons in the leading order,
so U → U† symmetry is broken as well. But the combination of P0 and U → U†,
which is the true parity, is not broken. In summary, the WZW term breaks both P0
and U → U† symmetries but leaves the combination of those unbroken.
For the WZW term including external sources, we refer to [24, 25] and quote their result here: S[U, l, r]W ZW = Nc Z D d5y ω − iNc 48π2 Z d4x µναβ(W (U, l, r)µναβ− W (1, l, r)µναβ) (2.60) where W is defined as W (U, l, r)µναβ = Tr(U lµlνlαU†rβ+ 1 4U lµU † rνU lαU†rβ + iU ∂µlνlαU†rβ + i∂µrνU lαU†rβ− iΣLµlνU†rαU lβ + ΣLµU † ∂νrαU lβ − ΣL µΣLνU † rαU lβ + ΣLµlν∂αlβ+ ΣµL∂νlαlβ − iΣLµlνlαlβ +1 2Σ L µlνΣLαlβ − iΣLµΣ L νΣ L αlβ) − (L ↔ R) , (2.61)
with
ΣLµ := U†∂µU , ΣRµ := U ∂µU†. (2.62)
Next we will go to order O(p6). We will not present the full O(p6) terms [26]. The only
term relevant to our calculation of AVV at order O(p6) is written as [10, 26]:
cuf fµναβhuµ{∇γf+γν, f αβ
+ }i . (2.63)
The above term provides only a small correction to the decay of π0 → γγ. Without
the WZW term, it would be the dominant contribution and the π0 would live much
longer [14].
2.3
A brief introduction to resonance chiral theory
As an effective field theory, ChPT can only describe 8 pseudoscalar mesons at very low energies E 1 GeV, and their interaction with electroweak gauge bosons, where E typically refers to the center of mass energy of the external states. At higher energies, ChPT will diverge because physically, other hadronic states (e.g. ρ meson) will be excited18. On the other hand, we know perturbative QCD works very well at very high energies E 1 GeV. Hence there exists a gap between ChPT and perturbative QCD at the intermediate energies19, where we lack an appropriate effective field theory. As we’ve explained in subsection 2.2.1, a necessary condition to construct an effective Lagrangian is that there exists a mass gap but there is no such gap in the intermediate energy range. If one wants to understand the intermediate energy range of QCD, we need some models which can provide some insights for the processes in the intermediate energy range. The model we will introduce next is resonance chiral theory.
It can be shown that if one writes down the most general Lagrangian that is con-sistent with the assumed symmetry, then for a given order in perturbation theory, the S-matrix amplitude will satisfy analyticity, perturbativity, unitarity and clus-ter decomposition [28]. But what symmetry and perturbation expansion parameclus-ter should one use at the intermediate energies if we want to incorporate not only Gold-stone bosons but also heavier pseudoscalars (P), scalars (S), vector mesons (V) and axial-vector mesons (A)? The large Nc expansion provides us some ideas.
2.3.1 Large Nc limit and its symmetry group
In large NcQCD one generalizes the symmetry group of SU (3)c to SU (Nc) in which
Nc → ∞. The generalization proposed by Gerard ’t Hooft [29] has many properties
that are shared with real QCD. One regards g to be of the order O(√1
Nc) when
Nc→ ∞, which is easy to understand from Eq. (2.13), because Nccontributes to the
18In practice, one notices that for a physical process (eg. ππ scattering), when E ∼ 4πF
0∼ 1 GeV,
the loop correction will be as important as the leading order diagram, which means ChPT fails at this energy scale.
19
denominator of Eq. (2.13). We will list some important results of large NcQCD here
[30]:
1) At leading order in N1
c, infinitely many stable mesons with zero width will
con-tribute to the correlators of quark bilinears.
2) The flavor group of the theory is U (Nf)L ⊗ U (Nf)R due to the absence of the
axial anomaly in the large Nc limit. So the symmetry is spontaneously broken to
U (Nf)V.
3) Meson loops are highly suppressed, so the dominant contributions to any process only come from tree-level diagrams.
To see why there isn’t an axial anomaly in the large Nc limit, we consider the
diver-gence of the Noether current J5µ:= ¯qγµγ5q corresponding to the U (1)A symmetry in
the chiral limit:
∂µJ5µ= Nfg2 16π2F a µνF˜ µν a . (2.64)
Now we multiply the numerator and denominator both by Nc, which leads to:
∂µJ5µ =
Nfg2Nc
16π2N c
Fµνa F˜aµν. (2.65)
Eq.(2.65) shows that in the large Nc limit where g2Nc is kept fixed, the denominator
goes to infinity, meaning that the axial current J5 is conserved and consequently the
chiral anomaly vanishes. This means that the chiral symmetry at large Ncis extended
to SU (3)V ⊗SU (3)A⊗U (1)A⊗U (1)V ' U (3)L⊗U (3)R. Thus, motivated by the large
Nc limit, Resonance Chiral Theory (RChT), which includes both Goldstone bosons
and other heavy meson states, possesses a U (3)L⊗ U (3)R symmetry. Concerning the
symmetry of the vacuum in the large Nclimit, the U (1)Atransformation does not leave
the quark condensate invariant, so the symmetry is broken down to SU (3)V ⊗ U (1)V
which is isomorphic to U (3)V. In contrast to ChPT where the symmetry group we
use was SU (3)L⊗ SU (3)R in Eq.(2.37), now in RChT the symmetry group becomes
U (3)L ⊗ U (3)R. In practice we are interested in massive resonances collected in
nonets R = R8⊕ R0. R8 transforms as an octet and R0 as a singlet under the group
SU (3)V: R8 → h(φ)R8h(φ)†, R0 → R0, (2.66) where R8 = P8 a=1R aλ
a/2 with λa being the Gell-Mann matrices.
We define the resonance kinetic terms as LRR,kin having the following form [31, 32,
33] LRR,kin = − 1 2h∇ µ Rµν∇αRανi + 1 4M 2 RhRµνRµνi + 1 2h∇ α R0∇αR 0 i − 1 2M 2 R0hR 0 R0i , (2.67) where Rµν stands for vector meson fields and axial-vector meson fields, R0 stands for
scalar and pseudoscalar. The above expressions are of order O(N0
to organize our Lagrangian. The first option for us is to organize it in terms of the number of resonance fields:
LRChT = LGB + LRR,kin+ LR+ L2R+ L3R+ · · · , (2.68)
where the interaction LR contains one resonance field, L2R two resonance fields and
so on. But organizing the Lagrangian in such way is not very useful when we inte-grate out the heavy resonances and try to match our results to ChPT at very low energies (E 1 GeV) because at very low energies we use chiral power counting. We therefore need to attribute chiral orders to the interaction terms, which is based on their contribution to the saturation of the LECs. Note that this is only a way to organize the Lagrangian in Eq. (2.68). There does not exist any expansion parameter in RChT except N1
c, so the expansion in chiral orders only makes sense at very low
energies where one can match to ChPT.
The resonance field R are at the order O(p2) [33]. The most general Lagrangian terms of order O(p4) are:
L(4)R =cdhSuµuµi + cmhSχ+i + idmhpχ−i + i dm0 NF hP ihχ−i + FA 2√2hAµνf µν − i + FV 2√2hVµνf µν + i + iGV 2√2hVµν[u µ , uν]i . (2.69)
An explicit form of a vector meson multiplet is
Vµν = 1 √ 2π 0+√1 6ω8+ 1 √ 3ω0 ρ + K∗+ ρ− −√1 2ρ 0+√1 6ω8+ 1 √ 3ω0 K ∗0 K∗− K¯∗0 −√2 6ω8+ 1 √ 3ω0 µν . (2.70)
The transformation behavior of the fields under parity and charge conjugation can be summarized as the following table:
Field names Fields Parity Charge conjugation Vector field Vµν (µ)(ν)Vµν −VµνT
Axial-vector field Aµν −(µ)(ν)Aµν ATµν
Scalar field S S ST
Pseudoscalar field P −P PT
Goldstone boson field uµ −(µ)uµ uTµ
Goldstone boson field hµν −(µ)(ν)hµν hTµν
External field χ± ±χ± χT±
External field f±µν ±(µ)(ν)f±µν ∓f±µνT
Table 1: Parity and charge conjugation transformation properties of the fields [31]. (0) = 1, (1) = (2) = (3) = −1 .
2.3.2 The RChT Lagrangian for the saturation of AVV at O(p6)
What we want to achieve is the most general RChT Lagrangian for the calculation of the AVV correlator at order O(p6). To construct such a Lagrangian the resonance
fields should include axial-vector (A) and vector meson fields (V) together with pion fields. We want to allow for several multiplets with the same quantum numbers (chan-nels). We refer to [33] for the Lagrangians for one multiplet per channel that saturate the LECs up to O(p6). However, we noticed that a plain addition of resonance
La-grangians from [33] is not sufficient to obtain a complete Lagrangian of order O(p6). It is because interactions between different vector meson multiples are also possible. We can perform the following replacements in the relevant Lagrangian terms in the reference [33] to find the Lagrangian terms which contain the maximal number of resonance fields:
V, f+ → Vi, Vj i, j = 1, 2, ..., n
A, f− → Ak, Al k, l = 1, 2, ..., n
where i, j, k, l refer to different resonance multiplets. For example, for the term εµναβhf+µν{Vαρ, Aβσ}igρσ, we will have the following replacement:
εµναβhf+µν{V αρ , Aβσ}igρσ → εµναβhViµν{V αρ j , A βσ l }igρσ
where i, j, l = 1, 2, 3, ..., n. In Tab. 2 we listed all the interaction terms20 between
vector mesons and axial-vector mesons together with those between vector mesons and pion fields. Except that the cAV V 2
ljk terms are of order O(p8), all the rest terms
in Tab. 2 are of order O(p6). We keep the terms cAV V 2
ljk because we want to estimate
the impact of higher order terms. For the definition related to the pion fields in chV V
j,k and cuV V 1j,k , cuV V 2j,k , we refer to Eq. (2.38). To obtain the terms that contain a
coupling constant interaction term origin cAV V 1 ljk εµναβhVjµν{V αρ k , A βσ l }igρσ κ V 11, κV A5 cAV V 2ljk εµναβhV µν j {∇ρV αρ k , ∇σA βσ l }i — chV Vjk εµναβhVjµν{V αρ k , h βσ}ig ρσ κV12 cuV V 1 jk εµναβhVjµν{∇αV βσ k , uσ}i κV16 cuV V 2jk εµναβhV µν j {∇σVkασ, u β}i κV 17 (cuV V 3 jk ) εµναβh{∇σVjµν, Vkασ}uβi κV V3
Table 2: Interaction terms containing maximal number of resonances
smaller number of resonance fields, we only need to make the resonance fields very heavy εµναβhViµν{V αρ j , A βσ l }igρσ Vj, Alheavy =======⇒ #εµναβhViµν{f αρ + , f βσ − }igρσ.
Therefore, by making Vj and Al very heavy, we can recover the results for the
εµναβhViµν{f αρ + , f
βσ
− }igρσ term up to a constant number which can be merged to the
coupling constant.
20cuV V 3
jk is redundant because it can be written as the linear combination −c uV V 2 jk − 1 2c hV V jk − 1 2c f vv1 jk .
3
AVV correlator and related quantities
“Physics is really nothing more than a search for ultimate simplicity.”
Bill Bryson
3.1
General decomposition
The flavor axial-vector—vector—vector (AVV) quark current correlator is given by [10]: Wabc µνρ(q1, q2) := i Z d4x1d4x2ei(q1·x1+q2·x2)h0|T{jµV a(x1) jνV b(x2) jρAc(0)}|0i =: 1 2d abcW µνρ(q1, q2) (3.1)
with the symmetric structure coefficients dabc introduced in Eq. (A.4).
And the vector (jV a
µ ) and axial-vector (jµAc) quark flavor currents are defined as:
jµV a:= ¯qγµ λa 2 q , j Ac µ := ¯qγµγ5 λc 2 q , (3.2)
respectively. In Eq. (3.1) q1 and q2 denote the outgoing momenta related to the
two vector currents. We denote the incoming momentum related to the axial-vector current by qA := q1+ q2. Below we will introduce scalar functions that depend on q12,
q2 2 and
qA2 = (q1 + q2)2. (3.3)
In practice, we will restrict our attention to a, b, c ∈ {3, 8} because these jV a
µ couple
to photons. The vector currents21 are conserved:
∂µjµV a = 0 , for a = 3, 8 . (3.4)
Because of current conservation (3.4), the structure Wµνρ satisfies the Ward
identi-ties
q1µWµνρ(q1, q2) = 0 = q2νWµνρ(q1, q2) . (3.5)
Because the two vector currents are indistinguishable, the Bose symmetry implies: Wµνρ(q1, q2) = Wνµρ(q2, q1) . (3.6)
21The eight vector currents jV a
µ are conserved in the chiral limit. In the limit where ms= md=
mu, the current conservation ∂µjµV a= 0 still holds eactly. In real QCD, only jV,3µ jµV,8are conserved
It’s worth to explore a general expression for the flavor AVV correlator which satisfies the Ward identity. We notice that in Eq. (3.1) the appearance of the axial current means there must be a Levi-Civita tensor µνρσ. The other two building blocks are q1
and q2. There will be 2 possibilities for the Levi-Civita tensor µνρσ with which the
momenta contract. The first possibility is that µνρσ covers all the indices of Wµνρ.
This gives
G1µνρ := µνρσq1σ, (3.7)
G2µνρ := µνρσq2σ. (3.8)
The second possibility is that µνρσ covers 2 indices. This gives
˜ Γ1µνρ := µναβq1αq β 2(q1+ q2)ρ= cµν(q1+ q2)ρ, (3.9) ˜ Γ2µνρ := µναβqα1q β 2(q1− q2)ρ= cµν(q1− q2)ρ, (3.10) ˜ Γ3µνρ := µραβq1αq β 2q1ν− νραβqα1q β 2q2µ = cµρq1ν − cνρq2µ, (3.11) ˜ Γ4µνρ := µραβqα1q β 2q1ν+ νραβqα1q β 2q2µ = cµρq1ν + cνρq2µ (3.12)
where we have used the definition cµν := µναβq1αq β 2.
We can also define:
˜
Γ5µνρ := cµρq2ν (3.13)
˜
Γ6µνρ := cνρq1µ. (3.14)
However, this set is overcomplete for two reasons. One can use the Schouten identity [34]
εµναβgστ = εσναβgµτ + εµσαβgντ + εµνσβgατ + εµνασgβτ (3.15)
to express ˜Γ5µνρ and ˜Γ6µνρ in terms of the other building blocks. One gets:
˜ Γ5 µνρ= 1 2(˜Γ 4 µνρ− ˜Γ 3 µνρ) − 1 2(˜Γ 1 µνρ− ˜Γ 2 µνρ) − q1· q2G2µνρ+ q 2 2G 1 µνρ, ˜ Γ6 µνρ= 1 2(˜Γ 4 µνρ+ ˜Γ3µνρ) − 1 2(˜Γ 1 µνρ+ ˜Γ2µνρ) − q1· q2G1µνρ+ q21G2µνρ. (3.16)
In addition, not all of the building blocks satisfy current conservation in the form of (3.5). Here the BTT (Bardeen, Tung, Tarrach) [35, 36] construction comes into the game. Our strategy is to use a combination of Eq. (3.7)-Eq. (3.12) such that the combinations of ˜Γj, G1 and G2 satisfy the current conservation conditions.
Γjµνρ = ˜Γjµνρ+ Aj1G1µνρ+ Aj2G2µνρ (3.17) where Aj1, Aj2 are22 the unknown coefficients with j = 1, 2, 3, 4. They can be solved from the Ward identities.
Finally, we give the four explicit expressions for the basis: Γ1µνρ := ˜Γ1µνρ = Cµν(q1+ q2)ρ, Γ2µνρ := ˜Γ2µνρ = Cµν(q1− q2)ρ, Γ3µνρ := ˜Γ3µνρ− (q1· q2)(G1µνρ− G 2 µνρ) = Cµρq1ν− Cνρq2µ− (q1 · q2) εµνρσ(qσ1 − q σ 2) , Γ4µνρ := ˜Γ4µνρ− (q1· q2) (G1µνρ+ G 2 µνρ) = Cµρq1ν+ Cνρq2µ− (q1· q2) εµνρσ(q1σ+ q σ 2) . (3.18)
Note that they have been chosen such that each of them has a well-defined behav-ior under exchange of {q1, µ} ↔ {q2, ν}. Finally, this leads us to the
decomposi-tion23: Wµνρ(q1, q2) = − 1 8π2 4 X j=1 Hj(q21, q 2 2, q 2 A) Γ j µνρ (3.19)
where we recall the relation (3.3). The coefficient in front of the sum in (3.19) is just a convention. It facilitates the comparison to [10]. The four scalar quantities Hj(q21, q22, q2A) are free from kinematical constraints and allow for dispersive
represen-tations in terms of their arguments. Bose symmetry implies: H1,3(q12, q 2 2, q 2 A) = +H1,3(q22, q 2 1, q 2 A) , H2,4(q12, q 2 2, q 2 A) = −H2,4(q22, q 2 1, q 2 A) . (3.20)
It is useful to translate our constraint-free scalar functions Hj to the scalar functions
introduced in [10] and vice versa. We find
H1 = −wL− q2 1 − q22 q2 A w(−)T +q 2 1+ q22− qA2 q2 A w(+)T , (3.21) H2 = w (−) T , (3.22) H3 = wT(+), (3.23) H4 = we (−) T , (3.24) wL = −H1− q21− q2 2 q2 A H2+ q21+ q22− q2 A q2 A H3. (3.25)
In the chiral limit, the longitudinal part of the AVV correlator, wL, is completely
fixed by the chiral anomaly [10]
wL= −
2Nc
q2 A
, (3.26)
i.e. wL does not depend on q12 and q22 and has a pole at qA2 = 0.
23An interesting physics argument to motivate that there must exist four form factors for the AVV
3.2
Transition form factors
We can define the transition amplitude of an axial-vector meson a decaying into an on-shell vector meson v and an off-shell photon γ∗:
hγ∗v| S |ai . (3.27)
In the first-order expansion in QED, it is given as
(2π)4δ4(q1+ q2− qa) × hγ∗(q1)v(q2)| (−ie)T {
Z
d4xJemµ (x)Aphotonµ (x)} |a(qa)i
= −ie(2π)4δ4(q1+ q2− qa) ∗µ(q1)
Z
d4x eiq1·xhv(q
2)| Jemµ (x) |a(qa)i
(3.28)
where we have defined
Jemµ (x) := ¯q(x) ˆQγµq(x) , q = (u, d, s)T , Q =ˆ 1
3diag(2, −1, −1) , (3.29) and Aphotonµ (x) to be the photon field. We can define the tensor Mµντa→γ∗v such that it
satisfies:
∗ν,v(q2)Mµντa→γ∗vτ,a(qa) :=
Z
d4x eiq1·xhv(q
2)| Jemµ (x) |a(qa)i (3.30)
with ∗ν,v(q2) and τ,a(qa) denoting the polarization of the (outgoing) vector meson v
and the (incoming) axial-vector meson a, respectively. Mµντa→γ∗v contains the QCD
dynamical information of a → γ∗v. In section 5.1, we will use the quark counting rules [37] to find the high-energy behavior of hγ∗v| S |ai in the Breit frame.
To relate Mµντa→γ∗vto the AVV correlator Wµντ one can use LSZ type formulae24:
Mµντa→γ∗v ∼ lim q2 2→m2v (q22− m2 v) lim q2 A→m2a (q2A− m2 a) W µντ (3.31)
We can define the TFFs HAV
j (Q2) as: HjAV(Q2) := lim q2 2→m2v (q22− m2v) lim q2 A→m2a (q2A− m2a) Hj(−Q2, q22, q 2 A) (3.32)
where j = 2, 3, 4 because the longitudinal Γ1
µνρ defined in Eq. (3.18) does not
con-tribute. From the above relation, it’s obvious that
Ma→γµνρ ∗v ∼
4
X
j=2
HjAV(Q2)Γjµνρ. (3.33) The high-energy behavior of the transition amplitudes will in turn put constraints on HAV
j (Q2) calculated by our hadronic model. We will discuss this in section 5.1 and
chapter 6.
The transition amplitude of a → γ∗γ∗ can be defined as:
hγ∗γ∗| S |ai . (3.34)
We now want to relate the scattering amplitude a → γ∗γ∗ to the AVV correlator similar to what we did for the a → vγ∗ transition. We can use perturbative QED to calculate the matrix element defined in Eq. (3.34). There are two outgoing virtual photons, so we need to expand the S matrix to second order in e:
(2π)4δ4(q1+ q2− qa)× hγ∗(q1)γ∗(q2)| 1 2(−ie) 2 T { Z d4xJemµ (x)Aphotonµ (x) Z
d4yJemν (y)Aphotonν (y)} |ai = (2π)4δ4(q1+ q2 − qa) × 1 2(−ie) 2 Z d4x Z d4y hγ∗(q1)γ∗(q2)| T {Jemµ (x)A photon µ (x)J ν em(y)A photon ν (y)} |ai = i(2π)4δ4(q1+ q2 − qa)e2∗µ(q1)∗ν(q2) i Z d4x Z d4y eiq1·xeiq2·yh0| T {Jµ em(x)J ν em(y)} |ai | {z } Mµντa→γ∗γ∗τ (3.35) where in the second equality we perform Wick contractions which cancel the 12 factor. As indicated in the above equation, we can define the tensor Mµντa→γ∗γ∗ that contains
the QCD information of a → γ∗γ∗ transition in the following way: Mµντa→γ∗γ∗τ,a:= i Z d4x Z d4y eiq1·xeiq2·yh0| T {Jµ em(x)J ν em(y)} |ai . (3.36)
To relate Mµντa→γ∗γ∗ to the AVV correlator Wµντ one can again use LSZ type formulae,
but this time we will be careful about the proportionality constants because in sec-tion 5.2 we will calculate the helicity amplitudes of the axial-vector meson a01(1260) decaying to two off-shell photons explicitly.
The neutral a01(1260) has the same quantum numbers as the axial-vector iso-vector quark current j3
Aµ := 12uγ¯ µγ5u − 1
2dγ¯ µγ5d. So the tensor M µντ
a→γ∗γ∗ is given by the
LSZ-type of formula [38] Mµντa0 1(1260)→γ∗γ∗ := lim q2 a→m2a0 1 i(q2 a− m2a0 1) √ 2Fa0 1ma01 × Z d4x Z d4y eiq1·xeiq2·y√2 h0| T {Jµ em(x)J ν em(y)j τ,3 A (0)} |0i , (3.37) where Fa0
1 and ma01 are the decay width and the mass of a
0
1, respectively. We add an
extra factor√2 in the integral to normalize jAτ,3. The electromagnetic current vector Jµ em can be decomposed as Jemµ = jVµ,3+√1 3j µ,8 V . (3.38)
